Fourier expansions of GL(3) Eisenstein series for congruence subgroups

by Deniz Balakci
Doctoral thesis
Date of Examination:2015-08-10
Date of issue:2016-04-14
Advisor:Prof. Dr. Valentin Blomer
Referee:Prof. Dr. Valentin Blomer
Referee:Prof. Dr. Jörg Brüdern
Persistent Address: http://dx.doi.org/10.53846/goediss-5610

 

 

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Abstract

English

In this thesis the Fourier expansions of all types of GL(3) Eisenstein series for the congruence subgroup Gamma_{0}(N) of SL(3)(Z) with N squarefree, are explicitly calculated. Further certain invariance properties of the Fourier coefficients are proved from which the functional equation can be deduced. This is explicitly carried out for the Eisenstein series twisted with a constant Maass form of prime level. In detail we treat the minimal Eisenstein series associated to the minimal parabolic subgroup, the Eisenstein series twisted by a Maass cusp form associated to the two maximal parabolic subgroups and the one twisted by a constant Maass form associated to the two maximal parabolic subgroups. These are the Eisenstein series which contribute to the spectral decomposition of the Hilbertspace of square integrable functions on the corresponding symmetric space. The Fourier expansions are calculated by first explicitly determining a set of coset representatives in Iwasawa decomposition for the summation sets in the definition of each Eisenstein series. Then the Fourier coefficients are calculated in terms of Whittaker functions. It turns out that the Fourier coefficients are a product of Whittaker functions with certain Dirichlet series, which split into a number theoretic part, the L-function of the Maass cusp form in the twisted case, divisor sums in the other cases, and into a combinatorial part. For the Eisenstein series twisted with a constant Maass form of prime level a scattering matrix is explicitly calculated and a transformation law of the Eisenstein vector is proven.
Keywords: Fourier expansions; GL(3) Eisenstein series; congruence subgroups
 
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